To acquire a latent that embeds both skill and knowledge, we will first define a compositional representation for any combination of sub-tasks in a multi-task learning environment.
A compositional representation is an embedding which encodes structural relationships between the items in the space [1]}.
The multi-task environment will help provide the disentanglement by allowing tasks and environments to be mixed in different combinations.
Consider a compositional task embedding \(\vec{v}\)  which encodes a set of tasks as the sum of the compositional embeddings for all subtasks.
To avoid enforcing a particular ordering for the completion of these subtasks, our planning space is built with commutativity, i.e. A+B = B+A.
Given this definition, the embedding of all tasks that have yet to be accomplished can be calculated as \((\vec{v}-\vec{u})\)  where \(\vec{u}\)  is the embedding vector for the tasks accomplished so far.
As we focus on semi-supervised machine learning, we don't specify the exact end state for the agent.
Instead, the policy \(\pi (a_t|O_{t},\vec{v}-\vec{u})\)  produces the action \(a_t\)  based on the current state \(O_{t}\)  and the “to do" task embedding.
